Remarks on the outer length billiards

This paper advances the study of outer length billiards by proving 3- and 4-periodic versions of the Ivrii conjecture and demonstrating the existence of functional spaces of billiard tables with invariant curves of $n$-periodic points, including an explicit geometric parameterization for the centrally symmetric $n=4$ case.

The Gist

Imagine you have a smooth, perfectly round stone (an "oval") sitting on a flat table. Now, imagine you are playing a game of billiards, but with a twist: instead of hitting a ball inside the stone, you are hitting a ball outside of it.

This is the world of Outer Length Billiards.

In this paper, mathematicians Misha Bialy and Serge Tabachnikov explore the rules of this specific game. They ask two big questions:

1. The "Ivrii Conjecture": If you play this game forever, will you ever find a "perfect" pattern where the ball bounces in a triangle or a square and repeats the exact same path over and over? Or are such perfect patterns so rare they are practically invisible?
2. The "Magic Tables": Can we design a weirdly shaped stone (not just a circle) that forces the ball to always bounce in a perfect triangle or square, no matter where you start?

Here is a breakdown of their findings using simple analogies.

1. The Game Rules: The "Fourth Billiard"

There are four famous ways to play billiards with a stone shape:

• Inner Billiards: The ball bounces inside the stone (like a pool table).
• Outer Billiards: The ball bounces outside the stone, but the rule is about the area the path covers.
• Symplectic Billiards: Another variation involving area.
• Outer Length Billiards (The Star of this Paper): The ball bounces outside, but the rule is about the total length of the path. The ball tries to find a path that is either the shortest or longest possible loop around the stone.

Think of it like a dog running around a fence. The dog wants to run a loop that is the "perfect" length. The "billiard map" is the rule that tells the dog where to turn next to keep that length perfect.

2. The Big Discovery: Perfect Loops are Ghosts

The authors prove a version of the Ivrii Conjecture for this game.

The Analogy: Imagine you are trying to find a specific type of cloud in the sky that is shaped exactly like a triangle. You might find one here or there, but if you look at the whole sky, the "clouds" that are perfect triangles are so thin and rare that if you threw a dart at the sky, the chance of hitting one is zero.

The Math Result:

• For 3-sided loops (triangles) and 4-sided loops (quadrilaterals), the authors prove that "perfect" repeating paths are incredibly rare.
• They are like a "null set." If you have a whole room full of possible starting positions, the ones that lead to a perfect, repeating triangle or square take up zero space. You can't find a "patch" of starting points that all lead to the same perfect loop.
• Why? They used a mathematical tool called a "twist map." Imagine a rubber sheet that twists as you pull it. If you twist it enough, you can't have a whole flat patch of the sheet stay perfectly still; it has to stretch or warp. This proves that perfect loops can't form a solid block; they are just isolated points.

3. The "Magic Tables": Designing the Stone

If perfect loops are so rare, can we build a stone that forces them to happen?

The Analogy: Imagine a video game level designer who wants to create a level where the player always runs in a perfect square, no matter where they start. Usually, this is impossible. But the authors show that if you are clever enough, you can design a "Magic Stone" (a specific oval shape) that has a "hidden track" where the ball must run in a perfect square.

The Findings:

• For Triangles (n=3): You can design a whole "family" of stones that force the ball to run in perfect triangles. It's not just one stone; it's a whole functional space of them.
• For Squares (n=4): This is where they got really creative. They focused on stones that are centrally symmetric (if you spin them 180 degrees, they look the same, like an ellipse or a rectangle).
  • They discovered that for these stones, the 4-periodic paths are always parallelograms.
  • They created a "recipe" (a formula) to build these stones. You start with a simple function (like a wave) and use it to "sculpt" the stone.
  • The Radon Curve Connection: They compared this to "Radon curves," which are shapes that look like they are made of different curves stitched together. They showed that you can build these "Magic Stones" using a similar stitching technique.

4. The "Invisible" Proof

The paper offers three different ways to prove that perfect triangles are rare (the Ivrii Conjecture for n=3).

• Method 1 (The Twist): Using the rubber sheet analogy (twist maps) to show the geometry doesn't allow for a solid block of triangles.
• Method 2 (Pure Geometry): They looked at the "tangent points" (where the ball touches the stone). They showed that if you try to wiggle a perfect triangle slightly, the geometry forces the ball to move in a direction that breaks the triangle. It's like trying to balance a pencil on its tip; the slightest nudge makes it fall.
• Method 3 (The Distribution): They used a high-level geometric concept called "sub-Riemannian geometry." Imagine a maze where you can only move in certain directions. They proved that in this maze, you can't build a flat "floor" (a surface) where every point is a perfect triangle. The maze is too twisty.

Summary

This paper is a mix of rigid rules and creative construction:

1. The Bad News: In a generic game of Outer Length Billiards, you will almost never find a perfect, repeating triangle or square. They are mathematical ghosts.
2. The Good News: If you are a master architect, you can design a specific, custom-shaped stone that forces the ball to run in perfect squares (parallelograms). They even gave you the blueprint (the formula) to build it.

It's a beautiful demonstration of how mathematics can tell us what is impossible in nature, while simultaneously showing us how to engineer exceptions to those rules.

Technical Summary

Here is a detailed technical summary of the paper "Remarks on the Outer Length Billiards" by Misha Bialy and Serge Tabachnikov.

1. Problem Statement and Context

The paper investigates outer length billiards, a discrete-time dynamical system acting on the exterior of a plane oval (a closed, strictly convex, smooth curve).

• Definition: An $n$-periodic orbit is an $n$-gon circumscribed about the oval such that the perimeter is extremal.
• Context: This system is the "fourth" billiard model, distinct from:
  1. Inner length (Birkhoff) billiards (inscribed, extremal perimeter).
  2. Outer billiards (circumscribed, extremal area).
  3. Symplectic billiards (inscribed, extremal area).
• Core Questions:
  1. Ivrii Conjecture: Does the set of periodic points have an empty interior (or is it a null set)?
  2. Integrability/Invariant Curves: Do there exist billiard tables (other than ellipses) that possess invariant curves consisting entirely of $n$-periodic points?

2. Methodology

The authors employ a combination of symplectic geometry, sub-Riemannian geometry, and classical differential geometry.

A. Twist Map Analysis

• The outer length billiard map $T$ is established as a positive twist map on the phase cylinder.
• The authors compute the generating function $S = l_1 + l_2 - \text{arc length}$ using envelope coordinates (support function $p(\alpha)$).
• They prove that the second iteration ($T^2$) is also a positive twist map. This property is crucial for proving the emptiness of the interior of periodic sets for periods 3 and 4.

B. Sub-Riemannian Geometry Approach

• To address the existence of invariant curves, the authors model the space of convex $n$-gons ($P_n$) as a manifold.
• They define a distribution $D_n$ on the space of polygons with fixed perimeter. This distribution is spanned by vector fields $\xi_i$ representing infinitesimal rotations of polygon sides about specific tangency points (related to the billiard reflection law).
• The problem of finding billiard tables with $n$-periodic invariant curves is reduced to finding horizontal curves (integral curves of $D_n$) in the space of polygons that are invariant under cyclic permutations.
• They utilize the Hsu criterion for non-singularity of horizontal curves to prove the existence of functional spaces of such curves.

C. Geometric and Analytic Proofs

• For the 3-periodic case, the authors provide two alternative proofs:
  1. A purely geometric proof involving the deformation of triangles and the motion of excircle tangency points, leading to a contradiction regarding the convexity of the oval.
  2. An analytic proof using the commutators of the vector fields in the distribution $D_3$ to show non-integrability.

3. Key Contributions and Results

A. Proof of the Ivrii Conjecture for Periods 3 and 4

• Theorem 1: The sets of 3-periodic and 4-periodic orbits for outer length billiards have empty interior.
• Mechanism: The proof relies on the fact that both $T$ and $T^2$ are positive twist maps. If a disc of 4-periodic points existed, the twist property would force a contradiction in the orientation of tangent vectors under iteration and inversion.
• Generalization: The authors note that if $T, T^2, \dots, T^k$ are all positive twist maps, then periodic orbits of periods $2, \dots, 2k$ have empty interiors.

B. Existence of Functional Spaces of Invariant Curves

• Theorem 2: For every $n \geq 3$, there exists a functional space of outer length billiard tables (close to a circle) that possess invariant curves consisting entirely of $n$-periodic points.
• Method: By perturbing the circular billiard (which trivially has such curves), they show that the distribution $D_n$ is completely non-integrable with growth type $(n, 2n-1)$. This allows for the construction of horizontal closed curves in the space of polygons, which correspond to the desired billiard tables.

C. Explicit Parameterization for $n=4$ (Centrally Symmetric Case)

• Theorem 3: The authors explicitly parameterize centrally symmetric billiard tables possessing 4-periodic invariant curves.
• Structure: These tables are generated by a single differentiable function $f(x)$ satisfying specific symmetry conditions ($f(x+\pi/2) = -f(x)$).
• Geometric Construction: They describe a construction method analogous to Radon curves. Starting with a convex arc in the first quadrant defined by a support function $p(\alpha)$, the full curve is constructed by extending via specific reflection and symmetry rules.
• Example: The standard ellipse is recovered as a specific case where $f(x) = \cos(2t)\sin(2x)$.

D. Alternative Proofs for Period 3

• The paper provides two distinct proofs for the 3-periodic Ivrii conjecture:
  1. Geometric: Uses the behavior of excircle centers and tangency points under infinitesimal deformation to show that convexity is violated if an open set of 3-periodic orbits exists.
  2. Algebraic: Analyzes the Lie brackets of the distribution $D_3$ in the space of triangles, showing that the distribution is non-integrable and cannot support a horizontal surface (open set of orbits).

4. Significance

• Unification: The paper unifies the study of outer length billiards with other billiard models (Birkhoff, outer, symplectic) regarding the rigidity of periodic orbits and the existence of integrable systems.
• Rigidity vs. Flexibility: It establishes a dichotomy:
  • Rigidity: Periodic orbits of low periods (3 and 4) cannot form open sets (Ivrii conjecture holds), implying generic non-integrability.
  • Flexibility: Despite this rigidity, there is a vast "functional space" of non-trivial tables (not just ellipses) that possess specific invariant curves of periodic points. This contrasts with Birkhoff's conjecture, which posits that complete integrability (foliation by invariant curves for all periods) implies the table is an ellipse.
• New Constructions: The explicit parameterization of 4-periodic tables provides a concrete method to generate non-elliptical integrable-like systems, expanding the known landscape of billiard dynamics.
• Methodological Advancement: The application of sub-Riemannian geometry and the analysis of the growth type of distributions on polygon spaces offers a powerful new toolkit for studying billiard dynamics and integrability.

In summary, Bialy and Tabachnikov demonstrate that while outer length billiards are generally non-integrable (no open sets of periodic orbits), they admit a rich family of specific tables with invariant periodic curves, characterized by functional degrees of freedom and explicit geometric constructions.